Dieudonné行列式的赋值计算
Computing Valuations of the Dieudonné Determinants
摘要：
研究离散赋值斜域上矩阵的Dieudonne行列式的赋值问题.在合理的计算模型下，我们提出了两个算法的一类DVSF，称为分裂。我们的算法是Murota（1995）的组合松弛和Moriyama-Murota（2013）的矩阵扩展的扩展，这两个算法都是基于组合优化的。虽然我们的算法需要一个上界的输出，我们给出了估计的斜多项式矩阵的界，并表明估计是有效的斜多项式矩阵。我们考虑这个问题的两个应用。第一个问题是非交换加权Edmonds问题（nc-WEP），即计算符号为非交换的矩阵的Dieudonne行列式的阶。我们表明，所提出的算法减少了nc-WEP在多项式时间的未加权问题。特别是，我们证明了nc-WEP在有理域上是可解的时间多项式中的输入位长度。通过建立线性微分/差分方程解空间上的公式，我们也给出了线性时变系统自由度分析的应用。
英文摘要：
This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama--Murota (2013), both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices. We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonné determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations.
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